By Yun Long, Asaf Nachmias, Weiyang Ning, Yuval Peres

ISBN-10: 1470409100

ISBN-13: 9781470409104

The Swendsen-Wang dynamics is a Markov chain regularly occurring by means of physicists to pattern from the Boltzmann-Gibbs distribution of the Ising version. Cooper, Dyer, Frieze and Rue proved that at the whole graph Kn the blending time of the chain is at so much O( O n) for all non-critical temperatures. during this paper the authors exhibit that the blending time is Q (1) in excessive temperatures, Q (log n) in low temperatures and Q (n 1/4) at criticality. in addition they supply an higher certain of O(log n) for Swendsen-Wang dynamics for the q-state ferromagnetic Potts version on any tree of n vertices

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**Extra info for A Power Law of Order 1/4 for Critical Mean Field Swendsen-wang Dynamics**

**Example text**

4) t ||σPt − π||T V = ||(|G1 |, 0)P , π||T V . 2 it suﬃces to couple the chains (Yt , Zt ) and (Yt , Zt ) such that they meet with probability Ω(1) in time t = Θ(1). 2, we have B B 2 E(Xt+1 n ≤ δ E(Xt2 ) − n . )− 1−δ 1−δ 45 46 YUN LONG, ASAF NACHMIAS, WEIYANG NING, and YUVAL PERES Applying this inductively we get E(Xt2 ) − B n ≤ δ t E(X02 ) ≤ δ t n2 . 1−δ For t ≥ 2 logδ 18 ( 1c − 12 ) and large n, we have E(Xt2 ) ≤ 1 1 1 − 4 c 2 2 2 n . 5) P Xt ≥ 1 1 1 − n ≤ . 2 and Markov’s inequality, if Xt ∈ [0, ( 1c − 12 )n], then Xt+1 ∈ √ [0, A n] with probability at least 1/2 for some large constant A.

2 concludes the proof. 1. 3. For any two magnetization chains Xt and Yt , deﬁne Jt = Xt − Yt . e. 2) τ := min{t : signJt = signJ0 }. 3 immediately. 5. Let Xt and Yt be two independent magnetization SW chain with d X0 ≥ n3/4 and Y0 = π. There exists positive constants δ, K, A and h such that P τ ≤ Kn1/4 ; Xτ −1 , Yτ −1 ∈ [A−1 n3/4 , An3/4 ] ; Jτ −1 ≤ hn5/8 ≥ δ . 5 we will use the following results. 8. 6. The stationary distribution π of the modiﬁed magnetization chain satisﬁes lim π[a1 n3/4 , a2 n3/4 ] = n→∞ 1 Z for any constants a2 ≥ a1 ≥ 0 where Z = constant.

Thus 3 1 20 P(|C(v)| ≥ ) ≤ P(τ ≥ ) = P(τ = ∞) + P( ≤ τ < ∞) . 16 with T = = 20 m and get by the previous display that 3 8 P(|C(v)| ≥ m/20) ≤ 2 − 2 + O( 3 ) + C1 −7/2 m−3/2 e− m/4 3 8 = 2 − 2 + O( 3 ) , 3 as long as 3 m ≥ A log m for large enough A. 34) concludes the proof of the upper bound on E|C1 |. We turn to the proof of the lower bound on E|C1 |. Recall that at each record minimum of the process {Yt } we are starting the exploration of a new component. 35) |C1 | ≥ m − γ + τ . < 0} . 28 YUN LONG, ASAF NACHMIAS, WEIYANG NING, and YUVAL PERES Thus, in order to complete the proof we will provide an upper bound on Eγ and a lower bound on Eτ .